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Journal of Financial Economics 81 (2006) 143–173

On the role of arbitrageurs in rational markets$

Suleyman Basaka, Benjamin Croitorub,Ã

aLondon Business School and CEPR, Regent’s Park, London NW1 4SA, UK bMcGill University, 1001 Sherbrooke Street West, Montreal, Quebec, H3A 1G5 Canada

Received 30 September 2003; received in revised form 29 April 2004; accepted 11 November 2004

Available online 10 January 2006

Abstract

We investigate the role of ‘‘arbitrageurs,’’ who exploit price discrepancies between redundant

securities. Arbitrage opportunities arise endogenously in an economy populated by rational,

heterogeneous investors facing investment restrictions. We show that an arbitrageur alleviates these

restrictions and improves the transfer of risk amongst investors. When the arbitrageur behaves

noncompetitively, taking into account the price impact of his trades, he optimally limits the size of his

positions due to his decreasing marginal profits. When the arbitrageur is subject to marginrequirements and is endowed with capital from outside investors, the size of his trades and capital are

endogenously determined in equilibrium.

r 2005 Published by Elsevier B.V.

JEL classification: C60; D50; D90; G11; G12

Keywords: Arbitrage; Asset pricing; Margin requirements; Noncompetitive markets; Risk-sharing

ARTICLE IN PRESS

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0304-405X/$ - see front matter r 2005 Published by Elsevier B.V.

doi:10.1016/j.jfineco.2004.11.004

$We are grateful to an anonymous referee for valuable suggestions and to Viral Acharya, Domenico Cuoco,

Michael Gallmeyer, Wei Xiong, seminar participants at the London School of Economics, McGill University, the

University of Vienna, the University of Wisconsin-Madison, the 2003 GERAD Optimization Days, the Blaise

Pascal International Conference on Financial Modeling, the 2004 American Finance Association Meetings, the

2004 European Finance Association Meetings and the 2005 Isaac Newton Institute Workshop on Mathematical

Finance for their helpful comments. Financial support from the Economic and Social Research Council for the

first author and from the Fonds Que ´ becois de Recherche sur la Socie ´ te ´ et la Culture, the Institut de Finance

Mathe ´ matique de Montre ´ al and the Social Sciences and Humanities Research Council of Canada for the second

author is gratefully acknowledged. All errors are solely our responsibility.Ã

Corresponding author. Fax: +1 514 398 3876.E-mail address: [email protected] (B. Croitoru).

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1. Introduction

The presence of apparent inconsistencies in asset prices has long been well documented.

Over the years, various types of securities such as primes and scores, stock index futures,

closed-end funds have been found to consistently deviate from their no-arbitrage values.Recent examples include the deviations from put-call parity in options markets (Ofek

et al., 2004) and the paradoxical behavior of prices in some equity carve-outs ( Lamont and

Thaler, 2003). For example, in the case of 3Com and Palm in March of 2000, the market

value of 3Com, net of its holdings of Palm, became negative on Palm’s first day of trading.

Such mispricings clearly lead to opportunities for arbitrage profits, and not surprisingly,

there is ample evidence of market participants engaging in trades designed to reap these

profits. For example, one type of arbitrage transaction, stock index arbitrage, is estimated

by the NYSE (www.nyse.com) to typically make up 7% to 9% of the program trading

volume, which itself represents about one-third of total NYSE trading, and many other

instruments, such as bonds and derivatives, are also frequently involved in arbitrages.

Moreover, there is also considerable anecdotal evidence of some investors specializing in

such trades. As Shleifer (2000, Chapter 4) puts it, ‘‘commonly, arbitrage is conducted by

relatively few, highly specialized investors.’’ A key example of such specialization is that of

hedge funds, many of which focus on arbitrage strategies (Lhabitant, 2002). Their

economic role, however, is highly controversial (especially after the infamous collapse of

the Long Term Capital Management hedge fund in 1998), underscoring the importance of

a rigorous study of the impact of arbitrage.

A growing thread of the academic literature studies these phenomena. Much of this

literature (see Shleifer, 2000, for a survey) has postulated the presence of irrational noisetraders whose erratic behavior generates mispricings. In such work, the term arbitrageur

typically refers to a rational trader, as opposed to irrational noise traders.

In this paper, we employ a different approach that is little explored in the literature so

far. We present an equilibrium in which all market participants are rational. Instead of

resorting to irrational noise traders to generate arbitrage opportunities, in our model these

arise endogenously in equilibrium, following from the presence of heterogeneous rational

investors subject to investment restrictions such as leverage constraints or restrictions on

short sales.1 Irrespective of whether or not noise traders are present in actual financial

markets, our view is that there may exist different categories of rational market

participants (e.g., long-term pension funds and short-term hedge funds) whose interactionsare sufficiently rich in implications. Our arbitrageurs are specialized traders who only take

on riskless, costless arbitrage positions. Our goal is to study the equilibrium impact of the

presence of these arbitrageurs. More specifically, we consider an economy with two

heterogeneous, rational, risk-taking investors and we examine how the presence of an

arbitrageur affects the way in which risk is traded between the investors, as well as the

ensuing equilibrium prices and consumption allocations. The investors could represent

real-life retail investors or mutual funds, while the arbitrageur in our model could be an

ARTICLE IN PRESS

1Recent empirical and theoretical work (e.g., Duffie et al., 2002; Ofek et al., 2004) points to impediments to

short-selling comparable to our investment restrictions as a possible cause for the presence of arbitrageopportunities. The restrictions on short-selling in Duffie et al. (2002) can be thought of as stochastic constraints

that could be incorporated into our work without considerably affecting our main message. Lamont and Thaler

(2003) emphasize the role of short-sale constraints, a part of the restrictions we assume, in making mispricings

possible.

S. Basak, B. Croitoru / Journal of Financial Economics 81 (2006) 143–173144

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arbitrage desk at an investment bank or a hedge fund that specializes in arbitrage

strategies. Our setup is comparable to the main workhorse models in asset pricing ( Lucas,

1978; Cox, Ingersoll and Ross, 1985), which provides the added benefit of having well-

understood benchmarks for our results.

Since our main message is not dependent on the particular modeling setup employed, wefirst convey our insights in the simplest possible framework, namely, a single-period

economy with a finite set of states of nature and elementary securities (paying off in a

single state) available for trading, and redundant securities available in at least one state.

In this setting, ‘‘mispricing’’ means that securities with identical payoffs (here, elementary

securities paying off in the same state) trade at different prices, thereby generating a

riskless arbitrage opportunity. Without making any particular assumptions on the

investors, apart from the fact that their holdings of risky securities are constrained, we

show that the presence of an arbitrageur makes it possible for the investors to trade with

each other more and thus partially circumvent the investment restrictions.

The intuition is as follows. Suppose that one investor desires a much higher payoff than

the other. The first investor will then purchase securities from the latter investor. If the

discrepancy across investors is sufficiently large, the investment restrictions will kick in and

prevent investors from attaining their desired payoffs. At this point, a role for the

arbitrageur comes into play: The arbitrageur purchases securities from the investor who

desires the lower payoff, and converts the securities into a position in the redundant

security, which is then resold at a higher price to the other investor. The larger the

arbitrageur’s trades, the more he relieves the effects of the constraints. Thus, even though

he does not take on any risk, the arbitrageur facilitates risk-sharing amongst investors. The

difference between the arbitrageur’s ask and bid prices is the amount of the mispricing andthe source of his arbitrage profit.

To further explore the implications of the arbitrageur’s presence, we consider a richer

continuous-time setup in which investment opportunities include a risky stock, a riskless

bond, and a risky derivative financial security that is perfectly correlated with the stock.

Real-life examples that are particularly close to the derivative in our model include a stock

index futures contract, or a total return swap on a stock index. To make our model more

concrete, the latter example is developed in some length; however, our model could

accommodate any redundant, zero-net-supply risky security, and the actual nature of the

derivative does not significantly affect our results. For simplicity, the derivative is assumed

to have the same exposure to risk (as measured by volatility) as the stock. ‘‘Mispricing’’then refers to any discrepancy in the mean returns offered by the risky investments (stock

and derivative). Because these risky investments are redundant, such a pricing

inconsistency generates a riskless arbitrage opportunity that can be exploited by short-

selling the risky security with the lower mean return and investing the proceeds in the

other, generating a profit proportional to the difference in mean returns. For tractability,

our investors have logarithmic utility preferences. Moreover, the investors face two

investment restrictions, a leverage constraint that prevents them from taking very large

positions in the stock, and a restriction that limits short positions in the derivative. These

restrictions may represent margin regulations imposed by an exchange or an intermediary,

as well as collateral requirements imposed by a counterparty. To generate heterogeneityand trading activity, we assume that the investors disagree on the mean stock return; the

more optimistic investor desires a higher exposure to risk and accordingly buys risky assets

from the more pessimistic investor.

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Mispricing is shown to occur when the disagreement between investors is sufficiently

pronounced. Absent the mispricing, the more optimistic agent, who is prevented from

investing more in the stock by his leverage constraint, would demand more derivative than

what the pessimist can sell (without violating his short-sale limit on the derivative). The

mispricing, however, makes the derivative less attractively priced and thus reduces theoptimist’s demand to a level compatible with the pessimist’s constraints, i.e., allows

markets to clear.

Our intuition on the arbitrageur’s role from the single-period example readily extends to

the continuous-time case. When the disagreement is large enough for mispricing to occur,

the binding investment restrictions prevent the investors from trading as much as they

would like. To take advantage of the arbitrage opportunity that is available under

mispricing, the arbitrageur buys stock (from the pessimist) and sells derivatives (to the

optimist). This allows the pessimist to decrease his exposure to risk and the optimist to

increase his exposure to risk, which helps both investors achieve risk exposures closer to

those that they would optimally demand in the absence of investment restrictions. Thus, by

taking advantage of the arbitrage opportunity, the arbitrageur allows the investors to trade

more risk, effectively relieving the investors of their investment restrictions. The greater the

size of the arbitrageur’s trades, the greater this effect: As a result, the size of the mispricing

is decreasing in the arbitrageur’s position. We further demonstrate that the arbitrageur

plays a specific role that is different from that of an additional investor: While the

arbitrageur always improves risk-sharing among investors, a third investor could either

improve or impair it, depending on his beliefs.

For the greater part of the analysis, we assume that the arbitrageur behaves

competitively and faces a pre-specified position limit, which may be linked to a bank’scapital requirements or may simply represent an internal position limit. Then, under

mispricing, the arbitrageur always takes on the largest arbitrage position allowed by his

position limit. Hence, the size of his position is exogenous. To be able to endogenize the

arbitrageur’s position size, we provide some extensions of our basic model. First, we

consider the case in which the arbitrageur is noncompetitive, in that he takes into account

the impact of his trades on equilibrium prices so as to maximize his profit. In addition to

generating much richer arbitrageur behavior, the noncompetitive assumption may be more

realistic in the case of arbitrageurs that in real life are typically large, few in number,

specialized, and sophisticated. Due to his decreasing marginal profit, the noncompetitive

arbitrageur finds it optimal to limit the size of his trades. This makes the occurrence of mispricing more likely, because the noncompetitive arbitrageur will never trade enough

to fully ‘‘arbitrage away’’ the mispricing and make it disappear (driving his profits to zero).

While our main intuition on the role of the arbitrageur is still valid, the extent to which

the arbitrageur alleviates the investors’ constraints may be smaller than in the

competitive case.

Finally, we consider a case in which the (competitive) arbitrageur is endowed with

capital and is subject to margin requirements that limit the size of his position in

proportion to the total amount of his capital. His capital is owned and traded by the

investors. This variation appears more realistic than our basic model, and is also richer in

implications. Because the investors may now invest in arbitrage capital, in equilibrium thereturn on arbitrage must be consistent with other investment opportunities; this allows us

to explicitly solve for the unique amount of arbitrage capital that is consistent with

equilibrium. Our approach is consistent with recent empirical work on the profitability of

ARTICLE IN PRESSS. Basak, B. Croitoru / Journal of Financial Economics 81 (2006) 143–173146

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arbitrage activity (e.g., Mitchell et al., 2002), which suggests that market imperfections

severely limit the return on arbitrage, and may drive it to a level that does not dominate

other investment opportunities.

The setup of this paper builds on the work of Detemple and Murthy (1994, 1997), who

solve for equilibrium in the presence of heterogeneous agents, both with and withoutportfolio constraints, but not with redundant derivative securities. Our framework more

closely follows the production economy of Detemple and Murthy (1994), in which the

technology determines the stock price dynamics exogenously and much of the action is

picked up by the bond interest rate. This, along with logarithmic investor preferences,

leads to considerable tractability and closed-form solutions in our analysis. Basak and

Croitoru (2000) go one step further by endogenizing the arbitrage opportunity and show

that in the presence of heterogeneous constrained investors such as our own, the mispricing

and arbitrage opportunity will be present under a broad range of circumstances. None of

these papers, however, includes a specialized arbitrageur.

To our knowledge, little work exists that examines the impact of specialized arbitrage in

an equilibrium in which all agents are rational. Related to our work is Gromb and

Vayanos (2002). These authors’ goals are somewhat different from ours, however, in that

they focus on the issue of a competitive arbitrageur’s welfare impact. In order to do this,

they examine a different form of constraints, specifically, that of a segmented market for

the risky assets, where different investors hold different risky assets. Without the

arbitrageur there is no trade; in the presence of the arbitrageur, trade occurs and a Pareto

improvement takes place (albeit not necessarily in a Pareto-optimal way). This is consistent

with the arbitrageur’s role in our model, which allows more trade to take place between

investors. We would not expect to obtain such a clear welfare result, however, since in ourmodel there is trade even in the absence of an arbitrageur and thus our benchmark is much

more complex. Despite improved risk-sharing, the overall welfare effect of our arbitrageur

is ambiguous, because his presence may move prices in a way that can be unfavorable to

the investors (Section 4). Our model is also closer in spirit to the standard asset pricing

models (and thus easier to relate to observable economic variables or well-understood

benchmarks). Loewenstein and Willard (2000) provide an equilibrium in which an

arbitrageur (interpreted as a hedge fund) plays a similar role when investors are subject not

to portfolio constraints, but to uncertain timing for their consumption. Unlike in our

model, arbitrage trades in Loewenstein and Willard are long lived.

Somewhat related are works, including Brennan and Schwartz (1990) and Liu andLongstaff (2004), that study a constrained arbitrageur’s optimal policy given the arbitrage

opportunity is present. Liu and Longstaff demonstrate that an arbitrageur may optimally

underinvest in an arbitrage opportunity, known to vanish at some future time, by taking a

smaller position than margin requirements would allow. Finally, our analysis complements

an important and growing strand of the finance literature that is often known as the study

of ‘‘limits on arbitrage.’’ This approach, inaugurated by De Long et al. (1990), is

thoroughly surveyed in Shleifer (2000). In these papers, arbitrage opportunities are

generated by the presence of irrational noise traders, and are allowed to subsist in

equilibrium due to various market imperfections (e.g., portfolio constraints, transaction

costs, short-termism, and model risk). Papers in this area that are particularly related toour work include Xiong (2001), who studies the impact of arbitrageurs (‘‘convergence

traders’’) on volatility. Kyle and Xiong (2001) extend his model to the study of contagion

effects between the markets for two risky assets, only one of which is subject to noise trader


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risk. Attari and Mello (2002) focus on the effect of trading constraints on the arbitrageur’s

impact and conclude (as we do) that these may have radical effects. Unlike the majority of

the literature, they assume that arbitrageurs take into account how their own trades affect

prices, as we do in part of our analysis.

Our work also complements the growing literature on asset prices in the presence of heterogeneous beliefs and short-sale restrictions, including Harrison and Kreps (1978),

Detemple and Murthy (1997), and Scheinkman and Xiong (2003). In these models, an

asset price may exceed its fundamental value (its valuation based on investors’ own

marginal rates of substitution) by a quantity reflecting the investors’ anticipated

speculative gains. There, investors are willing to pay more than the fundamental value

of an asset since ownership of the asset gives them the right to resell the asset to another

investor at an even higher price in the future. Some of these models’ features are also

present in our economy, which contains a short-sale constrained stock and heterogeneous

beliefs (but also a derivative security and an arbitrageur, not present in these papers). In

the presence of the redundant derivative and investment restrictions, our setting

additionally leads to riskless arbitrage opportunities, enabling a study of the role of an

arbitrageur—our primary focus.

The rest of the paper is organized as follows. Section 2 provides our single-period, finite-

state example on the role of the arbitrageur. Section 3 introduces our continuous-time

setup, and Section 4 describes the equilibrium with a competitive arbitrageur. Sections 5

and 6 are devoted to extensions to the case of a noncompetitive arbitrageur, and

to an arbitrageur endowed with capital and subject to margin requirements, respectively.

Section 7 concludes, and the Appendix A provides all proofs.

2. The role of the arbitrageur: an example

To convey our main intuition on the role of the arbitrageur and show that our main

point does not depend on the modeling setup employed, we start with an example in a

simple, one-period framework. Portfolio decisions are made at time 0 and securities pay off

at time 1. We first consider an economy with two heterogeneous investors i ¼ 1; 2 and no

arbitrageur. Consider two zero-net-supply securities, S and P , with time-0 prices also

denoted by S and P , and with identical payoffs of one unit if state o occurs, and zero

otherwise. The only difference between S and P is how trading in each is constrained.

Denoting by ai j investor i ’s investment, measured in number of shares, in security j (where j ¼ S ; P ), we assume that both investors’ holdings are subject to the position limits

ai S pb; ai

P XÀ g; i ¼ 1; 2. (1)

For simplicity, we assume that S and P are the only two available securities paying off in

state o. Then, investor i ’s payoff if state o occurs is given by Ai ¼ ai S þ ai

P . The constraints

in (1) alone do not place any restrictions on the choice of Ai , because one could always use

S to take on unlimited short positions and P to take on unlimited long positions.

Nonetheless, only a limited set of payoffs are available to the investors in equilibrium: No

investor can go long more than g shares of P , because the other investor cannot act as

a counterparty without violating his constraint on short-sales of P . Hence, no investor canreceive a state-o payoff greater than g þ b. A similar argument shows that no investor can

take on an unlimited short position, because the other investor cannot provide the

necessary counterparty. Thus, in equilibrium, both investors’ state-o payoffs must satisfy


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the constraint

Àb À gpAi pb þ g; i ¼ 1; 2. (2)

In short, investors’ position limits prevent them from acting as a counterparty to each

other, and this limits the trades that are possible in equilibrium.We now examine how the feasible trades (i.e., such that (2) holds) are affected in the

presence of a third agent, an arbitrageur. The arbitrageur is assumed to only take on

riskless arbitrage positions to take advantage of any price difference between S and P .

Whenever P aS , securities that have identical payoffs trade at different prices, giving rise

to a riskless arbitrage opportunity. For example, when P 4S , an arbitrage position

consisting of going long one share of S and short one share of P provides a time-0 profit

equal to P À S 40 without any future cash flows, and hence no risk. Basak and Croitoru

(2000) show how such price discrepancies arise in equilibrium in the presence of

constrained heterogeneous investors. In our subsequent analysis, we provide equilibria in

which such mispricings also arise endogenously. For now, we take the presence of

mispricing ðP 4S Þ as given and address the issue of the economic role of an arbitrageur

who exploits the mispricing. It is straightforward to verify that, given the constraints (1),

the other direction of mispricing ðS 4P Þ cannot arise in equilibrium, as agents would then

add an unbounded arbitrage position to their portfolio. Furthermore, we take the size of

the arbitrageur’s position (a3S 40 shares of S , a3

P ¼ Àa3S shares of P ) as given. In the

absence of any constraint, an arbitrageur would optimally choose to hold an infinitely

large arbitrage position and thus equilibrium would be impossible. To avoid this, later

sections assume either that the arbitrageur is constrained or that he is non-price-taking.

Then, an investor who desires a large short position can short as many as b þ a3

S sharesof S (instead of only b if the arbitrageur were not present), because the arbitrageur acts as

an additional counterparty over and above the other investor. A similar argument applies

to large long positions, so that the constraint on investors’ state-o payoffs (the analogue of

(2)) is now

Àb À g À a3S pAi

pb þ g þ a3S ; i ¼ 1; 2. (3)

Eq. (3) reveals that the presence of the arbitrageur makes it possible for investors to trade

more. To see the intuition for this, take the example of investor 1 desiring to buy a large

payoff from investor 2 (possibly because the former is more optimistic, or less risk averse).

Without an arbitrageur, once investor 1 has purchased b shares of S and g shares of P ,trading between the two investors becomes ‘‘stuck’’ no matter how great the heterogeneity

between them. The arbitrageur makes it possible for investor 1 to buy an extra a3S shares of

P from him. This position is converted by the arbitrageur into a position in S , that is then

voluntarily shorted by investor 2, given the constraints were preventing him from shorting

more before the arrival of the arbitrageur. The net result is that the arbitrageur has made it

possible for the two investors to trade more risk with each other, even though he does not

take on any risk himself. The arbitrageur plays the role of a financial intermediary: The

larger his position, the more he alleviates the constraints.

It should be pointed that the role of the arbitrageur is specific. Adding a third investor

(who can take on risk), instead of an arbitrageur, to the economy can either worsen orrelieve the constraint in (2). In the presence of mispricing, the third investor will always

bind on one of his position limits (1); which one he binds on depends on his desired cash

flow. If this third investor (indexed by 3Ã) binds on his upper constraint on S , the


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constraint on investor i 2 1; 2’s state-o cash flow becomes

À2b À gpAi pb þ g À a3Ã

P . (4)

While the lower bound is lower than in (3), showing an alleviation of the lower constraint,

the upper bound can be either lower or higher, and accordingly the constraint either moreor less stringent. A similar point can be made when agent 3 Ã binds on his constraint on P .

The effect of a third investor is thus ambiguous, highlighting the specific role of an

arbitrageur, who always improves risk-sharing.

While this section deals with a simplistic economic setup, in Section 4 we show how the

exact same points can be made within a much more general continuous-time modeling

framework. It should also be noted that the analysis in this section is robust to changes in a

large number of assumptions: The preferences, beliefs, and endowments of investors 1 and

2, the type of portfolio constraints (position limits or constraints on portfolio weights,

whether they are one- or two-sided, whether they are homogeneous or heterogeneous

across agents), and the net supplies of the securities (meaning that our analysis is also

equally valid in production and exchange economies). All that is really needed is the

presence of redundant securities, portfolio constraints, and heterogeneity amongst

investors.

3. The economic setting

This section describes the basic economic setting, which is a variation on the Cox,

Ingersoll and Ross (1985) production economy. The economy is populated by two

investors, who trade so as to maximize the cumulated expected utility of consumption in astandard fashion, and an arbitrageur, who is constrained to hold only riskless arbitrage

positions. The description of the arbitrageur is relegated to later sections, as related

assumptions will be different in each section.

3.1. Investment opportunities

We consider a finite-horizon production economy. The production opportunities are

represented by a single stochastic linear technology whose only input is the consumption

good (the numeraire). The instantaneous return on the technology is given by

dS ðtÞ

S ðtÞ¼ mS dt þ sdwðtÞ, (5)

where mS and s ðs40Þ represent the constant mean return and volatility on the technology,

respectively, S is the amount of the good invested, and w is a one-dimensional Brownian

motion. For convenience, we often refer to the technology as the ‘‘stock’’ S .

The two investors observe the return of the stock, but have incomplete (though

symmetric) information on its dynamics. They know s (from the stock return’s quadratic

variation), but must estimate mS via its conditional expectation and rationally update their

beliefs in a Bayesian fashion with heterogeneous prior beliefs. We denote by mi S ðtÞ the

conditional estimate of mS by investor i ¼ 1; 2 at time t, given his prior beliefs and hisobservation of the stock’s realized return; by ¯ mðtÞ ¼ ðm1

S ðtÞ À m2S ðtÞÞ=s we denote

the normalized difference between the two investors’ estimates. The process ¯ m represents

the disagreement between the investors on their relative optimism or pessimism about the


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stock mean return. The process ¯ m is given exogenously, since all learning about mS comes

from observing the (exogenous) realized return on the stock. For simplicity, we shall

assume that ¯ m40, and we call investor 1 (2) the optimistic (pessimistic) investor. Consider

the well-known Gaussian example in which investor i ’s prior belief is normally distributed

with mean mi ð0Þ and variance vð0Þ. Then, investors’ estimates have dynamics dmi S ðtÞ ¼

ðvðtÞ=sÞ dwi ðtÞ, where vðtÞ ¼ vð0Þs2=ðvð0Þt þ s2Þ, implying d ¯ mðtÞ ¼ ÀðvðtÞ=sÞ ¯ mðtÞ dt, or ¯ mðtÞ ¼

¯ mð0Þ½s2=ðvð0Þt þ s2Þs. Assuming ¯ mð0Þ40 implies ¯ mðtÞ40, 8t. Our subsequent analysis goes

through for ¯ mo0, with the only (minor) difference being the presence of another mispricing

case arising in equilibrium (Sections 4–6), that mirrors the mispricing case arising for ¯ m40.

Finally, we let

wi ðtÞ ¼1

s

Z t

0

dS ðsÞ

S ðsÞÀ

Z t

0

mi S ðsÞ ds

!(6)

denote investor i ’s estimate of the Brownian motion w. By Girsanov’s theorem, w

i

is astandard Brownian motion under investor i ’s beliefs, and investor i ’s perceived stock

return dynamics are given by

dS ðtÞ

S ðtÞ¼ mi

S ðtÞ dt þ sdwi ðtÞ; i ¼ 1; 2. (7)

Though not a central feature of our model, heterogeneity in beliefs is employed to generate

trading activity between the logarithmic investors.

In addition to the stock, investors may invest in two zero-net-supply, non-dividend-

paying securities. One is an instantaneous riskless bond with return

dB ðtÞ

B ðtÞ¼ rðtÞ dt, (8)

and the other a risky derivative with perceived return process

dP ðtÞ

P ðtÞ¼ mi

P ðtÞ dt þ sdwi ðtÞ; i ¼ 1; 2. (9)

Similarly to the stock, investors observe the derivative return process but not its dynamics

coefficients, and hence they draw their own inferences about the derivative mean return mP .

Since the derivative pays no dividends, we take its volatility parameter s (assumed to equal

stock volatility for simplicity) to define the derivative contract P , as is standard in theliterature (e.g., Karatzas and Shreve, 1998). The interest rate r and derivative mean return

mi P , on the other hand, are determined endogenously in equilibrium. Any zero-net-supply

security is an example of this derivative. Real-life examples that are particularly close in

spirit to our idealized derivative include a stock index futures contract, or a total return

swap on a stock index; we take the latter as a leading example in our subsequent

discussion. In the case of the derivative being an overnight total return swap on the stock, a

long investor receives the total stock return and pays an instantaneously riskless swap rate

rP (possibly different from r), with each unit yielding between t and t þ dt:

dS ðtÞS ðtÞ

À rP ðtÞ dt ¼ ½mi S ðtÞ À rP ðtÞ dt þ sdwi ðtÞ; i ¼ 1; 2. (10)

All of our results, derived for the general case of a derivative with mean return mi P , are

equally valid in the case of the total return swap if we take rP ðtÞ ¼ rðtÞ þ mi S ðtÞ À mi

P ðtÞ. The


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generalization to the case of the derivative having a different volatility than the stock is

straightforward and is considered in an earlier version of this manuscript (Basak and

Croitoru, 2003). Apart from increased notational complexity, all our main results here are

robust to such a generalization. Moreover, defining the derivative by a dividend process

would not entail major changes to our analysis. All of the expressions in the paper wouldremain valid but the derivative volatility s would be endogenously determined via a

present value formula. Our main points and intuition would be unaffected.

3.2. Investors’ optimization problems

With all the uncertainty generated by a one-dimensional Brownian motion, the stock

and derivative returns are perfectly correlated, and so the derivative is redundant.

However, restrictions on investors’ risky investments generate an economic role for the

derivative. Letting yi ðyi

B ;yi

S ;yi

P Þ, with yi

B ðtÞ ¼ W i ðtÞ À yi

S ðtÞ À yi

P ðtÞ, denote the amounts

of investor i ’s wealth, W i , invested in B , S , and P , respectively, we assume that in addition

to short sales being prohibited, stock investments face leverage limits, and investments in

the derivative have limited short-selling restrictions at all times:

0pyi S ðtÞpbW i ðtÞ; yi

P ðtÞXÀ gW i ðtÞ, (11)

where bX1, g40. For retail investors, the stock investment restriction may be due to

margin requirements, while for mutual funds they may represent leverage restrictions. If

the derivative security is an over-the-counter contract, such as a total return swap, the

derivative investment constraint (11) may be due to a collateral requirement imposed by a

counterparty. If the derivative is an exchange-traded contract, such as a futures contract,then the restriction (11) may represent margin regulations imposed by the exchange. The

one-sided restriction on holdings of the derivative is for expositional simplicity, with

the two-sided case being a straightforward extension that does not affect our main

conclusions. In particular, suppose that the derivative holdings are additionally restricted

to satisfy the leverage restriction yi P ðtÞpgW i ðtÞ, g40. Then, there would be additional

scenarios in the investors’ optimal holdings of Proposition 3.1, leading to additional

equilibrium cases as discussed in Section 4.2.

The risky assets are exposed to the same amount of risk ðsdwi ðtÞÞ, and thus it is natural

to expect them to be priced accordingly and to offer identical mean returns. In fact, if this

were not the case, there would be an arbitrage opportunity available, exploited by short-selling the asset with the lower mean return and investing the proceeds in the other, which

would generate a riskless arbitrage profit proportional to the difference in mean returns.

This motivates our definition of mispricing between the stock and derivative under investor

i ’s beliefs as the difference between their mean returns,

Di S ;P ðtÞ mi

S ðtÞ À mi P ðtÞ. (12)

We note that the observed return agreement across investors enforces agreement on the

mispricing: D1S ;P ðtÞ ¼ D2

S ;P ðtÞ DS ;P ðtÞ. In the presence of investment restrictions, DS ;P need

not be zero. If DS ;P ðtÞ40, the stock has a higher expected return and we say that it is

‘‘cheap’’ relative to the derivative; conversely, if DS ;P ðtÞo0, the stock is more ‘‘expensive’’(and the derivative is cheaper). For example, if the derivative is the overnight total return

swap, the mispricing is given by any deviation of the swap rate from the interest rate since,

whenever rP ar, an arbitrage opportunity is available. If rP 4r, the swap is more expensive


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than the stock, and a strategy consisting of buying the stock, financing the purchase by

riskless borrowing, and taking a short position in the swap yields an instantaneous

arbitrage profit of $ðrP À rÞ per dollar in the stock; vice versa if rP or.

Investor i is endowed with initial wealth W i ð0Þ. He chooses consumption policy ci and

investment strategy yi so as to maximize his expected lifetime logarithmic utility subject toa dynamic budget constraint and position limits, that is, to solve the problem

maxci ; yi

E i Z T

0

logðci ðtÞÞ dt

!(13)

s. t. dW i ðtÞ ¼ ½W i ðtÞrðtÞ À ci ðtÞ dt þ fyi S ðtÞ½mi

S ðtÞ À rðtÞ þ yi P ðtÞ½mi

P ðtÞ À rðtÞg dt

þ ½yi S ðtÞ þ yi

P ðtÞsdwi ðtÞ ð14Þ

and ð11Þ.

This differs from the standard frictionless investor’s dynamic optimization problem

due to the investment restrictions, potential redundancy, and mispricing between the

risky investment opportunities S and P . The solution of investor i ’s problem is reported

in Proposition 3.1. Cases in which P is cheap are ignored, as they cannot arise in

equilibrium.

Proposition 3.1. Investor i ’s optimal consumption and investments in the stock S and

derivative P are given by

c

i

ðtÞ ¼

W i ðtÞ

T À t ; (15)

when there is no mispricing, i.e. DS ;P ðtÞ ¼ 0,

yi S ðtÞ

W i ðtÞ¼

2 ½0;b

0;

&yi

P ðtÞ

W i ðtÞ¼

XÀ gs; s.t.yi

S ðtÞ

W i ðtÞþ

yi P ðtÞ

W i ðtÞ¼mi

S ðtÞ À rðtÞ

s2(a)

if mi S ðtÞXrðtÞ À gs2

Àg if mi S ðtÞorðtÞ À gs2 (b)

8>>>><>>>>:

(16)

and when there is mispricing, with the stock being cheaper than the derivative, i.e. DS ;P ðtÞ4

0,

yi S ðtÞ

W i ðtÞ¼

b if mi S ðtÞ4mi

P ðtÞXrðtÞ À ðg À bÞs2

b if mi S ðtÞ4rðtÞ À ðg À bÞs2

4mi P ðtÞ

mi S ðtÞ À rðtÞ

s2þ g if rðtÞ À ðg À bÞs2Xmi

S ðtÞXrðtÞ À gs2

0 if rðtÞ À gs24mi

S ðtÞ;

8>>>>>><>>>>>>:

(c)

(d)

(e)

(f)

(17)

yi P ðtÞW i ðtÞ

¼

mi P ðtÞ À rðtÞ

s2À b if (c)

Àg if (d)

Àg if (e)

Àg if (f):

8>>>>><>>>>>:

(18)


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An investor can be in one of six possible scenarios, (a)–(f), depending on the economic

environment and whether there exists mispricing (cases (c)–(f)) or not (cases (a)–(b)). When

there is no mispricing and the perceived mean return is sufficiently low, the investor desires

the lowest possible exposure to risk and thus binds on his lower constraint on both risky

assets (case (b)). Otherwise, he is indifferent between all feasible investments in the tworedundant risky assets that lead to his optimal risk exposure (case (a)) (that is, investments

ensure that the volatility of his portfolio equals the market price of risk).

Under mispricing (cases (c)–(f)), a riskless, costless, profitable arbitrage opportunity is

available, and the investor indulges in it to the greatest extent possible as allowed by the

position limits. Effectively, the investor adds to his portfolio an arbitrage position

consisting of a long position in the cheap stock and a short position in the expensive

derivative. This implies that the investor is always binding on at least one of the investment

restrictions in (11), and thus uniquely determines the allocation between the stock and the

derivative. When the investor is relatively optimistic about the mean return of the stock

and the derivative is not too expensive (case (c)), he desires a high risk exposure, wishing to

go long in both risky assets, and so ends up binding on the leverage constraint on S . When

the derivative is more expensive (DS ;P 40 or, in the case of a total return swap, higher rP )

by a large amount, he is driven to his short sale limit on P ; if additionally the investor is

pessimistic about the stock, he desires a low risk exposure and thus does not invest in the

stock (case (f)), otherwise he goes long in the stock (cases (d)–(e)). As is demonstrated in

Section 4.2, mispricing cases (c) and (e) may occur in equilibrium (with or without an

arbitrageur). Cases (d) and (f) do not.

4. Equilibrium with a competitive arbitrageur

4.1. The arbitrageur

We assume that in addition to the two investors i ¼ 1; 2, the economy is populated by an

arbitrageur indexed by i ¼ 3. The arbitrageur is assumed to take on only riskless, costless

arbitrage positions, and to maximize his expected cumulated profits under a limit on the

size of his positions. This natural formulation of the arbitrageur is somewhat simplistic,

but it does capture the most important distinguishing characteristics of a specialized

arbitrageur such as an arbitrage desk in an investment bank or a hedge fund focusing on

arbitrage strategies. Denoting the arbitrageur’s (dollar) investments by y3

ðy3B ; y

3S ; y

3P Þ,

his problem can be expressed as:

maxy3

E 3Z T

0

C3ðtÞ dt

!(19)

s.t. C3ðtÞ dt ¼ y3S ðtÞm3

S ðtÞ þ y3P ðtÞm3

P ðtÞ þ y3B ðtÞrðtÞ

È Édt

þ ½y3S ðtÞ þ y3

P ðtÞsdw3ðtÞ, ð20Þ

y3S ðtÞ þ y3

P ðtÞ þ y3B ðtÞ ¼ 0, (21)

ðy3S ðtÞ þ y3

P ðtÞÞs ¼ 0, (22)

0py3S ðtÞpM . (23)


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An equivalent interpretation of the no-risk condition (22) is that the arbitrageur’s position

is uncorrelated with the stock return, as would ideally be the case for a hedge fund’s return.

The arbitrageur may be a trader of an investment bank or a hedge fund. For an investment

bank, the investment restriction (23) imposed on the arbitrage trader may be linked to the

capital requirements faced by the bank, with a credit limit controlled by the FederalReserve. In the case of a hedge fund, the investment restriction imposed on the arbitrageur

may simply represent an internal position limit.

Conditions (21)–(22) imply

y3P ðtÞ ¼ Ày3

S ðtÞ; y3B ðtÞ ¼ 0, (24)

so it follows that the arbitrageur’s holdings in all securities are pinned down by his stock

investment. Therefore, the constraint on holdings of the stock (23) also limits the

arbitrageur’s holdings of the other securities.

In the absence of mispricing ðDS ;P ¼ 0Þ, it is straightforward to verify that the

arbitrageur is indifferent between all portfolio holdings that satisfy (23)–(24), since they all

provide him with zero cash flows (and he can do no better than this). In the presence of

mispricing ðDS ;P 40Þ, however, as long as y3S 40, the arbitrageur receives a profit that is

proportional to y3S . The optimal policy then is to choose the maximal possible value for y3

S ,

namely M . From (24), the other holdings are given by y3P ðtÞ ¼ ÀM and y3

B ðtÞ ¼ 0, and the

arbitrageur’s instantaneous profit rate is given by

C3ðtÞ ¼ y3S ðtÞm3


P ðtÞ ¼ M DS ;P ðtÞ. (25)

For example, in the case of the derivative being a total return swap on the stock, the

arbitrageur may be purchasing the stock, financing the purchase by borrowing at theriskless rate and simultaneously entering the swap, whereby he pays the stock return and

receives the swap rate rP . In the absence of mispricing, the two interest rates are equal.

Under mispricing, the notional swap rate is higher than the interest rate (by DS ;P ), and the

difference generates the arbitrageur’s profit.

It should be pointed out that the price-taking arbitrageur’s solution, as well as the value

of his profits, is independent of his beliefs and preferences. Due to his constraints,

(21)–(22), the arbitrageur consumes his profits as they are made ðc3 ¼ C3Þ and holds no net

wealth, so that without loss of generality, he can be assumed to be risk neutral, without any

time preference. Any increasing utility function for the arbitrageur would lead to the same

behavior as for the risk-neutral arbitrageur, in this section as well as in Sections 5 and 6.

4.2. Analysis of equilibrium

We now proceed with the description of the equilibrium (defined, in standard fashion, as

the security prices such that agents’ optimal policies clear the security markets, given the

prices) in our economy. For convenience, quantities of interest are expressed as a function

of the aggregate wealth, W W 1 þ W 2, and the proportion of aggregate wealth held by

investor 1, l W 1=W . Several cases are possible in equilibrium, depending on investor 1

and 2’s binding constraints. We denote each case by the optimization case of each investor;

for example, in equilibrium ða; aÞ each investor is in case (a), in equilibrium ða; bÞ investor 1is in case (a) and investor 2 in case (b), etc.

Proposition 4.1 reports the possible equilibrium cases, details the conditions for their

occurrence, and characterizes economic quantities (prices and consumptions) in each case.


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Proposition 4.1. In equilibrium, the investors’ optimization cases, equilibrium mispricing and

interest rate, and distribution of wealth dynamics are as follows.

When

¯ mðtÞp gslðtÞ

þ min slðtÞ

; s1 À lðtÞ

ðb À 1Þ þ M lðtÞW ðtÞ

!& ', (26)

investors are in ða; aÞ and

DS ;P ðtÞ ¼ 0; rðtÞ ¼ lðtÞm1S ðtÞ þ ð1 À lðtÞÞm2

S ðtÞ À s2, (27)

and

dlðtÞ ¼ lðtÞð1 À lðtÞÞ2ð ¯ mðtÞÞ2 dt þ lðtÞð1 À lðtÞÞ ¯ mðtÞ dw1ðtÞ. (28)

When

¯ mðtÞ4 ðg þ 1ÞslðtÞand lðtÞX 1b 1 À M W ðtÞ

, (29)

investors are in ða; bÞ and

DS ;P ðtÞ ¼ 0; rðtÞ ¼ m1S ðtÞ À

g þ 1

lðtÞÀ g

!s2, (30)

and

dlðtÞ ¼ sðg þ 1Þð1 À lðtÞÞ2

lðtÞdt þ ð1 À lðtÞÞ dw1ðtÞ

!. (31)

When

¯ mðtÞ4gs

lðtÞþ

s

1 À lðtÞb À 1 þ

M

lðtÞW ðtÞ

and lðtÞo

1

b1 À

M

W ðtÞ

, (32)

investors are in ðc; eÞ and

DS ;P ðtÞ ¼ ¯ mðtÞs Àðb À 1Þs2

1 À lðtÞÀgs2

lðtÞÀ

M s2

lðtÞð1 À lðtÞÞW ðtÞ40, (33)

rðtÞ ¼ m2S ðtÞ À ð1 À gÞs2 þ ðb À 1Þ

lðtÞ

1 À lðtÞ

þM

ð1 À lðtÞÞW ðtÞ !s2, (34)

and

dlðtÞ ¼ lðtÞ DS ;P ðtÞ b À 1 þM

W ðtÞ

!þ ð1 À lðtÞÞð ¯ mðtÞ À DS ;P ðtÞ=sÞÂ Ã2

& 'dt

þ lðtÞð1 À lðtÞÞð ¯ mðtÞ À DS ;P ðtÞ=sÞ dw1ðtÞ. ð35Þ

In all cases, the aggregate wealth dynamics follow:

dW ðtÞ ¼ W ðtÞ m1S ðtÞ À

1

T À t

À M DS ;P ðtÞ

!dt þ W ðtÞsdw1ðtÞ (36)

and investor i ’s consumption and the arbitrageur’s profit are given by

c1ðtÞ ¼lðtÞW ðtÞ

T À t; c2ðtÞ ¼

ð1 À lðtÞÞW ðtÞ

T À tand C3ðtÞ ¼ M DS ;P ðtÞ. (37)


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Three cases are possible in equilibrium: ða; aÞ and ða; bÞ in which there is no mispricing

and the arbitrageur makes no profits, and ðc; eÞ in which mispricing occurs. We now

describe how, as the divergence in beliefs across investors grows, the equilibrium may move

into region ðc; eÞ, where the arbitrageur is active and affects the equilibrium.

In our economy, investors trade due to their disagreement on the mean stock return,with the optimist buying stock from the pessimist. If the disagreement is not too large

(equilibrium case ða; aÞ), then the optimist puts more than 100% of his wealth into stock

and the pessimist puts some fraction between 0% and 100% into stock. Since the

technology return determines the stock price, to clear the market, the interest rate is set at a

value depending on the average degree of optimism across investors (as in the framework

of Detemple and Murthy, 1994). As the optimist becomes even more optimistic, he buys

more and more stock from the pessimist, until either the optimist hits his leverage

constraint or the pessimist hits his no-short-sale constraint. Which constraint is hit first

depends on the distribution of wealth between the two investors, with the poorer one

having a tendency to hit his constraint first. Let us suppose that the pessimist hits his

constraint first. Then, as the optimist becomes even more optimistic beyond this point, he

buys derivatives from the pessimist, even though the pessimist has more stock that he could

sell to the optimist if the optimist were not leverage constrained. As the optimist becomes

even more optimistic, he eventually buys so many derivatives from the pessimist that the

pessimist hits his constraint on short derivative positions.

At this point, additional trades between the optimist and the pessimist would be

impossible, and risk-sharing would be ‘‘stuck,’’ were it not for the presence of the

arbitrageur. As the optimist becomes even more optimistic, the arbitrageur buys stock

from the pessimist (who still has some stock left because he did not hit his no-short-salesconstraint before the optimist hit his leverage constraint) and sells derivatives to the

optimist. The arbitrageur is essentially converting stock positions into derivative positions.

As the optimist continues to become more optimistic, he buys more and more derivative,

until a point is reached at which the arbitrageur hits his position limit M or the pessimist

sells the arbitrageur all of his stock. Let us assume that the arbitrageur’s position limit is

hit first, which occurs when the economy moves from region ða; aÞ into region ðc; eÞ, and the

derivative becomes mispriced relative to the stock. As the optimist becomes more

optimistic beyond this point, the arbitrageur cannot sell him more derivative. Instead, the

derivative’s mean return is reduced and the optimist limits the amount of derivative that he

buys because it becomes more expensive.Returning to the case of the derivative being a total return swap, a positive mispricing

means that the notional riskless rate used to price total return swaps becomes higher than

the actual riskless rate in the economy. This is the opposite of what happens when there is a

‘‘repo squeeze,’’ and the notional swap rate is less than the market interest rate. This higher

notional interest rate makes the total return swaps more costly to the optimist, and entices

him to reduce his holding to the level the arbitrageur is allowed by his position limit to sell

to the optimist. On the other side of the book, however, the arbitrageur has a plentiful

supply of stock for sale by the pessimistic investor. Thus, for markets to clear, the riskless

rate needs to drop below the swap rate; this induces the pessimist to reduce the amount of

stock he wants to sell to the amount the arbitrageur can buy (M ).If, on the other hand, the pessimistic investor is much wealthier than the optimist, as

divergence in beliefs grows the optimist hits his leverage constraint on the stock before the

pessimist hits his constraint on the derivative. If this is the case, market clearing does not


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require that the optimist’s demand of the derivative be reduced, because the pessimist is

wealthy enough to act as a counterparty and sell to the optimist derivatives without hitting

his constraint; thus, there is no mispricing in equilibrium (case ða; bÞ).

In short, the arbitrageur is active under high divergence in beliefs across investors and

provided the pessimist is not much wealthier than the optimist. The arbitrageur buys stockfrom the pessimist and sells derivatives to the optimist. Under mispricing (equilibrium case

ðc; eÞ), the derivative is more expensive than the stock, and the arbitrageur profits from the

difference. If we also assume that the investors face a leverage constraint on investments in

the derivative ðyi P ðtÞpgW i ðtÞ; g40Þ, an additional case may occur in equilibrium, in which

there is no arbitrage opportunity, and hence no role for the arbitrageur. In that case, the

leverage constraint prevents the optimist from purchasing more derivatives than what

the pessimist can short, and so the equilibrium does not require the derivative to become

less attractive to the optimist via the mispricing. The optimist is prevented by the

additional leverage constraint from purchasing derivatives from the arbitrageur, who in

turn cannot trade. Our analysis of region ðc; eÞ remains unchanged, however, and our

main message on the role of the arbitrageur is thus unaffected by this generalization of our

basic setup.

Henceforth we focus on the analysis of region ðc; eÞ, where the arbitrageur has an

economic impact and so affects economic quantities. In region ða; aÞ, the equilibrium is as

in an unconstrained economy as analyzed by Detemple and Murthy (1994), because no

constraints are binding; in region ða; bÞ, the equilibrium is similar to a constrained

economy without a derivative as in Detemple and Murthy (1997). In both of these cases,

there is no arbitrage opportunity available. It is straightforward to verify that region ðc; eÞ

does occur for plausible parameter values: For example, according to (32), if g ¼ 0:5,b ¼ 1:5, s ¼ 0:1, M =W ðtÞ ¼ 0:1, and lðtÞ ¼ 0:5, region ðc; eÞ occurs whenever investor

1’s estimate of the mean return of the stock exceeds investor 2’s estimate by at least

2.4% per year. In our analysis, we focus on the terms reflecting the effects of the

arbitrageur’s presence and the size of his position (measured by M ). All of our analyses

and comparisons are valid only for given aggregate wealth ðW Þ and distribution of wealth

across investors ðlÞ. For comparison with our economy, we introduce the following two

benchmarks:

Economy I: no constraints, no derivative; otherwise, as our economy.

Economy II: no arbitrageur; otherwise, as our economy.

In Economy I, all equilibrium quantities are as in our region ða; aÞ, while in Economy II,they are as in our economy for the particular case M ¼ 0.

The value of the mispricing can be interpreted as the amount of heterogeneity in beliefs

that investors cannot trade on due to the binding constraints. Whenever no constraints are

binding and thus risk-sharing is not limited, the mispricing is always equal to zero.

However, on moving from region ða; aÞ to region ðc; eÞ, the mispricing starts at a value of

zero and then increases linearly with heterogeneity ð ¯ mÞ while interagent transfers remain

stuck due to the constraints. The mispricing is decreased by the presence of the arbitrageur,

and decreases further as the size of his position increases; compared with our benchmarks,

we have DIS ;P ¼ 0oDS ;P oD

IIS ;P . Keeping in mind our interpretation of the mispricing as the

amount of untraded heterogeneity, this suggests an improvement in risk-sharing due to thearbitrageur’s presence.

The value of the interest rate is increased by the presence of the arbitrageur; we have

rI4r4rII. This is indirect evidence of an improvement in risk-sharing between investors


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due to the arbitrageur’s presence: improved risk-sharing decreases investors’ precautionary

savings. For the bond market to clear in spite of this, it is necessary for the interest rate to

rise, thus becoming closer to its value in an unconstrained economy. Note that, as in the

production economy of Detemple and Murthy (1994), since the technology determines the

stock price dynamics exogenously, it is the interest rate that has to adjust for markets toclear and thus it picks up much of the action. The interest rate can be solved for explicitly

because investor demands are myopic and linear in the stock’s Sharpe ratio (i.e., entirely

determined by the interest rate).

In short, the impact of the arbitrageur on all price dynamics parameters is to make them

closer to their values in an unconstrained economy; the larger his position, the closer the

resulting equilibrium is to an unconstrained equilibrium. The conditions for region ðc; eÞ to

occur (and the constraints to bind) are also affected. From (32), it is clear that the larger

the arbitrageur’s position, the less likely it is for region ðc; eÞ to occur and the constraints to

bind, perturbing risk-sharing between investors. In all cases, the effect of increasing the size

of the arbitrageur’s position (M ) on the expressions is tantamount to an increase in b or g

(i.e., an alleviation of the constraints). All of this suggests that the arbitrageur alleviates the

effect of the portfolio constraints. The overall effect of his presence on investor welfare is

ambiguous, however, because his trades move prices in a way that can hurt the investors.

The arbitrageur’s trades reduce the mean stock excess return (above the riskless rate),

which hurts both investors, but increase the mean derivative return, benefiting the

optimistic investor (who is long) and hurting the pessimist. The increase in the interest rate

benefits the pessimist (long in the bond), but hurts the optimist. Thus, in spite of the

unambiguous improvement in risk-sharing due to the arbitrageur’s trades, the welfare

impact of the arbitrageur’s presence is ambiguous. However, note that all agents in ourmodel are better off than in a no-trade economy, as in Gromb and Vayanos (2002),

because they could choose to not trade.

Finally, interpreting the leverage limit parameter b and the limited short-selling

parameter g as policy variables set either by an exchange or a regulatory body, we may

investigate the impact of leverage and short-selling limits on the arbitrage activity. From

the characterization in Proposition 4.1, we see that increasing these limits ðg;bÞ shrinks the

mispricing region, and hence makes mispricing less likely to occur. Furthermore,

increasing these limits increases the interest rate, moving it closer to the perfect risk-

sharing case, and reduces the magnitude of the mispricing and thus the arbitrageur’s

profits: Overall, there is less arbitrage activity. This is because increasing these limitsalleviates the risky investment restrictions on the investors, and hence the two investors are

better able to share risk (with or without the arbitrageur). This in turn provides for a

smaller role for the arbitrageur, reducing the mispricing and the arbitrageur’s profits.

4.3. The arbitrageur and risk-sharing

To clarify the role of the arbitrageur, we quantify the transfers of risk that take place in

region ðc; eÞ, as well as in our benchmark economies I and II. We will additionally consider

the following benchmark case:Economy III: no securities (the only investment opportunity is the stock).

In Economy III, investors do not trade and they invest all of their wealth in the stock

ðyi S ¼ W i Þ.


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We introduce Fi , investor i ’s measure of risk exposure, defined as the proportion of his

wealth held in risky investments,

Fi ðtÞ yi

S ðtÞ þ yi P ðtÞ

W i ðtÞ. (38)

In the absence of trade (Economy III), all of agent i ’s wealth is invested in the stock, and so

Fi III ¼ 1 for both investors. When trade is possible, heterogeneity in beliefs leads the more

optimistic investor to optimally take on a higher risk exposure than in the no-trade case,

and the more pessimistic agent to take on a lower risk exposure. Corollary 4.1 reports the

investors’ equilibrium risk exposures both in the absence of binding constraints

(benchmark economy I and region ða; aÞ of our economy) and in the equilibrium case

ðc; eÞ of our economy.

Corollary 4.1. In the absence of binding constraints, equilibrium risk exposures are

F1I ðtÞ ¼ 1 þ ð1 À lðtÞÞ

¯ mðtÞ

sand F2

I ðtÞ ¼ 1 À lðtÞ¯ mðtÞ

s. (39)

In the ðc; eÞ equilibrium,

F1ðtÞ ¼ b þ g1 À lðtÞ

lðtÞþ

M

lðtÞW ðtÞand

F2ðtÞ ¼ 1 À g À ðb À 1ÞlðtÞ

1 À lðtÞÀ

M

ð1 À lðtÞÞW ðtÞ. ð40Þ

In the presence of constraints, once the disagreement ð ¯ mÞ is sufficiently large for theconstraints to bind, the amount of risk that is traded becomes ‘‘stuck’’: Portfolio holdings

do not depend on ¯ m any more. It is straightforward to verify that, assuming the conditions

for ðc; eÞ to occur in our economy are fulfilled,

F1IIIðtÞoF1

IIðtÞoF1ðtÞoF1I ðtÞ; and F2

IIIðtÞ4F2IIðtÞ4F2ðtÞ4F2

I ðtÞ. (41)

Thus, the presence of the arbitrageur makes the allocation of risk closer to what it would

be without constraints. A natural measure of the amount of risk that is shared is the

difference F1 À F2, which is equal to zero in the no-trade case (benchmark Economy III)

and, in the absence of binding constraints (benchmark economy I and region ða; aÞ in oureconomy), grows linearly with heterogeneity (it is then equal to ¯ m=s). In our equilibrium

ðc; eÞ,

F1ðtÞ À F2ðtÞ ¼b À 1

1 À lðtÞþ

g

lðtÞþ

M

W ðtÞlðtÞð1 À lðtÞÞ, (42)

revealing how the amount of risk that is shared between the investors grows with the size of

the arbitrageur’s position. The mechanism through which the arbitrageur facilitates risk-

sharing between the investors is the one already described in Section 2: The arbitrageur

effectively acts as a financial intermediary who acts as an additional counterparty to the

investors. By converting stock positions into derivative positions, the arbitrageur allowsthe investors to trade more with each other. The analysis in Section 2 can be repeated with

only insubstantial changes, provided one replaces investors’ state-o cashflows (Ai s) in

Section 2 with their risk exposures (Fi s). In the absence of the arbitrageur (benchmark


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Economy II), investors’ risk exposures are

F1IIðtÞ ¼ b þ g

1 À lðtÞ

lðtÞand F2

IIðtÞ ¼ 1 À g À ðb À 1ÞlðtÞ

1 À lðtÞ. (43)

Even though the optimistic investor 1 would like to increase his risk exposure (to F1I ) by

increasing his holding of the derivative (since he is already leverage-constrained on the

stock), this is impossible because then investor 2 would not be able to act as a counterparty

due to his lower constraint on P . By shorting the derivative, the arbitrageur provides an

extra counterparty, which allows for an increase in F1 (by M =lW ) that is proportional to

the arbitrageur’s position limit M . Similarly, it is impossible for the pessimistic investor 2

to invest as little in the stock as he optimally would, because then not all of the stock would

be held and so markets would not clear. By investing in the stock, the arbitrageur makes it

possible for the pessimistic investor to hold less of it and thereby reduce his risk exposure

(by M =½ð1 À lÞW ). The introduction of the arbitrageur in the economy is equivalent to

that of an extra, zero-net-supply security (e.g., a total return swap on the stock), in which

both investors face a position limit proportional to M . This security is purchased from the

pessimistic investor by the innovator, the arbitrageur, and resold to the optimistic investor

at a higher price (or a higher notional riskless rate, in the case of the swap). The difference,

the mispricing, makes up the arbitrageur’s profit.

Finally, we note that much of the above analysis is not dependent on the logarithmic

utility assumption. The investors’ portfolio holdings in region ðc; eÞ follow directly from the

binding constraints and the market clearing conditions. Thus, whenever mispricing occurs,

these holdings obtain for arbitrary investor preferences or beliefs, and our analysis of the

arbitrageur’s contribution to improved risk-sharing is equally valid. Only the occurrence of mispricing is needed for this to follow. Basak and Croitoru (2000, Proposition 5) show how

and when mispricing may arise in equilibrium under very general assumptions,

demonstrating the robustness of our analysis.

4.4. Comparison with an economy with three investors and no arbitrageur

It is well known that, ceteris paribus, increasing the number of agents typically improves

risk-sharing in an economy.2 Thus, naturally the question arises as to whether adding a

third (risk-taking) investor to our economy would improve risk-sharing in the same way

that the arbitrageur does. Proposition 4.2 presents an example of equilibrium that can arisein this case. We index the third investor (logarithmic expected utility maximizer) by i ¼ 3Ã

and, accordingly, denote his wealth by W 3Ã and his estimate of mS by m3ÃS . As before,

W W 1 þ W 2 and l W 1=W .

Proposition 4.2. In the presence of a third optimizing agent ð3ÃÞ, but no arbitrageur, if

¯ mðtÞ4sðb À 1Þ1 þ

W 3ÃðtÞW ðtÞ

1 À lðtÞþ gs

1 þW 3ÃðtÞ

W ðtÞ

lðtÞ þW 3ÃðtÞ

W ðtÞ

þm1

S ðtÞ À m3ÃS ðtÞ

s

W 3ÃðtÞ

W ðtÞ, (44)

ARTICLE IN PRESS

2For example, in a standard economy the risk aversion of the representative agent A satisfies 1=A ¼ Si ð1=Ai Þ

and so decreases when the number of agents in the economy increases, ceteris paribus. Hence, the market price of

risk, Asd, where sd is the volatility of aggregate consumption, and the individual consumption volatilities, Asd=Ai ,

also decrease.

S. Basak, B. Croitoru / Journal of Financial Economics 81 (2006) 143–173 161

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Àgs 1 þW ðtÞ

W 3ÃðtÞ

pm1


spgs

1 þ W 3ÃðtÞW ðtÞ

lðtÞ, (45)

and

1 À lðtÞ

lðtÞ þ W 3ÃðtÞW ðtÞ

Xb À 1, (46)

then in equilibrium investors are in optimization cases ðc; e; cÞ and the risk exposures of

investors 1 and 2 are

F1ÃðtÞ ¼ bs þ gs1 À lðtÞ

lðtÞ þ W 3ðtÞW ðtÞ

þm1


s

W 3ÃðtÞ

lðtÞW ðtÞ þ W 3ÃðtÞ, (47)

and

F2ÃðtÞ ¼ sð1 À gÞ À ðb À 1ÞslðtÞ þ W 3ÃðtÞ

W ðtÞ

1 À lðtÞ. (48)

Investors’ equilibrium risk exposures reveal that a third risk-taking investor affects risk-

sharing differently than the arbitrageur does. While F2IoF

2ÃoF2

II, implying that the

presence of the extra investor 3Ã makes investor 2’s risk exposure closer to its value in an

unconstrained economy ðF2I Þ, the effect on investor 1 is ambiguous. Depending on investor

3Ã’s beliefs, F1Ã can be either higher or lower than if the third investor were not present.

Accordingly, the effect of investor 3Ã on the amount of risk shared between investors 1 and

2 is ambiguous. Our measure of risk-sharing is now equal to

F1ÃðtÞ À F2ÃðtÞ ¼ ðb À 1Þs1 þ

W 3ÃðtÞW ðtÞ

1 À lðtÞþ gs

1 þW 3ÃðtÞ

W ðtÞ

lðtÞ þ W 3ÃðtÞW ðtÞ

þm1


s

1

1 þ lðtÞ W ðtÞ

W 3ÃðtÞ

. ð49Þ

Compared to the value of this measure in the absence of binding constraints (which

remains equal to ¯ m, whether or not a third investor is present), if investor 3 Ã is sufficiently

optimistic (while still allowing for the conditions in the proposition to hold), risk-sharingcan be degraded by his presence. An example of plausible parameters for which this

happens is as follows: g ¼ 0:5, b ¼ 1:5, sS ¼ sP ¼ 0:1, lðtÞ ¼ 0:5, W 3ÃðtÞ=W ðtÞ ¼ 0:4,

m1S ðtÞ ¼ 0:1, m2

S ðtÞ ¼ 0:05, and m3ÃS ðtÞ ¼ 0:105. Then, the equilibrium is as described in

Proposition 4.2, and the amount of risk shared between investors 1 and 2 is smaller than if

the third investor were not present: F1 À F2 is equal to 0.1761 versus 0.2 (and 0.5 in an

unconstrained economy, whether or not a third investor is present).

Thus, our arbitrageur, whose presence always improves risk-sharing across investors (as

demonstrated by (42)), performs a specific economic function.

5. Equilibrium with a noncompetitive arbitrageur

In this section, we consider another imperfection for the financial markets, in that the

arbitrageur has market power therein. This could naturally arise in markets with limited


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liquidity or in specialized markets with few participants. There is considerable evidence

(see, e.g., Attari and Mello, 2002) suggesting that in such markets, large traders ‘‘move’’

prices.

When the arbitrageur has market power in the securities markets, he takes account of

the price impact of his trades on the level of mispricing across the risky assets. Thisconsideration is shown to induce much richer arbitrageur trades than those in the

competitive case, in which under mispricing, the arbitrageur simply takes on the largest

position allowed by his investment restriction. The importance of this noncompetitive case

is further underscored by the fact that trading restrictions on the arbitrageur are no longer

needed to bound arbitrage, and hence attain equilibrium with an active arbitrageur, as

demonstrated in this section.

5.1. The noncompetitive arbitrageur

For tractability, we consider a myopic arbitrageur in the sense that he maximizes his

current profits (as opposed to lifetime profits). This shortsightedness may be interpreted as

capturing in reduced form arbitrageurs that are either short-lived or have short-termism, as

often observed in practice. This may, for example, be due to the separation of ‘‘brains and

resources’’ that prevails in real-life professional arbitrage (Shleifer, 2000, Chapter 4): If the

arbitrageur reports to shareholders who are less sophisticated than himself, he may be

enticed to choose short-term profits over a long-term strategy that may return less in the

short run.

An arbitrageur with market power faces a problem that is different—and more

complex—than the competitive arbitrageur of Section 4. The noncompetitive arbitrageurobserves the investors’ demands as functions of prices (as determined in Section 3), and

then chooses the size of his own trades so as to maximize his concurrent profits by taking

the exact counterposition of the investors’ trades so that markets clear. Of course, he will

only be active when there is mispricing and hence positive profits to be made; otherwise his

profits are zero. Accordingly, given the investors’ optimal investments (Proposition 3.1),

clearing in the security markets implies the following when mispricing (case ðc; eÞ) occurs:


1 À lðtÞÀgs2

lðtÞÀ

y3S ðtÞs2

lðtÞð1 À lðtÞÞW ðtÞ. (50)

Hence, the arbitrageur’s influence on security prices manifests itself, via (50), as the size of his stock position ðy3

S Þ that affects mispricing. The larger his position, the lower the

mispricing—in a sense, the prices move against him: The value of his marginal profit (the

mispricing) is reduced by his trades.

The noncompetitive arbitrageur solves the following optimization problem at time t:

maxy3

S ðtÞ;DS ;P ðtÞ

C3ðtÞ ¼ y3S ðtÞDS ;P ðtÞ (51)

s.t. y3S ðtÞ þ y3

P ðtÞ þ y3B ðtÞ ¼ 0; ðy3

S ðtÞ þ y3P ðtÞÞ s ¼ 0; (21)2(22)

and DS ;P ðtÞ satisfies (50).The arbitrageur solves simultaneously for the optimal size of his arbitrage position and the

equilibrium mispricing, given his understanding of how the equilibrium mispricing

responds to his trades (50). As for the competitive arbitrageur, the costless, riskless


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arbitrage conditions, (21)–(22), uniquely determine the noncompetitive arbitrageur’s

holdings in financial securities in terms of his stock investment. Finally, we note that the

maximization problem (51) is a simple, concave quadratic problem.

5.2. Analysis of equilibrium

Proposition 5.1 reports the possible equilibrium cases and the characterization for each

case.

Proposition 5.1. The equilibrium with a noncompetitive arbitrageur is as follows.

When ¯ mðtÞpgs=lðtÞ þ min s=lðtÞ; ðb À 1Þs=1 À lðtÞÈ É

; investors are in ða; aÞ, and when

¯ mðtÞ4ðg þ 1Þs=lðtÞ and lðtÞX1=b, investors are in ða; bÞ. In both cases, DS ;P ðtÞ, rðtÞ, and lðtÞ

are as in the competitive arbitrageur equilibrium described in Proposition 4.1.

When ¯ mðtÞ4gs=lðtÞ þ sðb À 1Þ=1 À lðtÞ and lðtÞo1=b; investors are in ðc; eÞ and the

arbitrageur’s equilibrium stock investment and profit are given by

y3S ðtÞ ¼

lðtÞð1 À lðtÞÞW ðtÞ

2s¯ mðtÞ À

sðb À 1Þ

1 À lðtÞÀ

gs

lðtÞ

!(52)

and

C3ðtÞ ¼lðtÞð1 À lðtÞÞW ðtÞ

4¯ mðtÞ À

sðb À 1Þ

1 À lðtÞÀ

gs

lðtÞ

!2

, (53)

respectively, and the equilibrium mispricing, interest rate, and distribution of wealth dynamics

are given by

DS ;P ðtÞ ¼1

2¯ mðtÞs À

ðb À 1Þs2

1 À lðtÞÀgs2

lðtÞ

!, (54)

rðtÞ ¼ m2S ðtÞ À s2 þ s ¯ mðtÞ

lðtÞ

2þ

1

2gs2 þ

1

2

lðtÞ

1 À lðtÞs2ðb À 1Þ, (55)

and

dlðtÞ ¼ lðtÞ DS ;P ðtÞ ðb À 1Þ þy3

S ðtÞ

W ðtÞ

!þ ½ð1 À lðtÞÞð ¯ mðtÞ À DS ;P ðtÞ=sÞ2

& 'dt

þ lðtÞð1 À lðtÞÞð ¯ mðtÞ À DS ;P ðtÞ=sÞ dw1ðtÞ, ð56Þ

respectively. In all cases, the aggregate wealth dynamics follow:

dW ðtÞ ¼ W ðtÞ m1S ðtÞ À

1

T À t

À y3

S ðtÞDS ;P ðtÞ

!dt þ W ðtÞsdw1ðtÞ. (57)

As in the competitive case, three cases are possible in equilibrium, with case ðc; eÞ being

the mispricing case. Comparison with the competitive equilibrium of Proposition 4.1

reveals that the region of mispricing is larger in the noncompetitive equilibrium. Hence, the

presence of the noncompetitive arbitrageur makes mispricing more likely to occur. This is

intuitive: in the presence of mispricing, the competitive arbitrageur always trades to the fullextent allowed by the position limit, and thus is more likely to fully ‘‘arbitrage away’’ the

mispricing, making it disappear. In the noncompetitive case, in contrast, he limits his

trades so as to make the mispricing (and positive arbitrage profits) subsist under a broader


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range of conditions. It is impossible for the arbitrageur to make the mispricing case more

likely because, whenever he trades, he improves risk-sharing across investors and reduces

the mispricing. As a result, unlike in the competitive case, the conditions for regions ðc; eÞ

are as if the arbitrageur were not present.

As compared with the no-arbitrageur economy (Economy II, Proposition 4.1 withM ¼ 0), we see that the equilibrium mispricing is reduced by a half, DS ;P ðtÞ ¼ ð1=2ÞDII

S ;P ðtÞ.

That is, instead of providing a perfect counterparty to the two investors (which would

eliminate the mispricing), when the arbitrageur is a non-price-taker, he provides only one-

half of a perfect counterparty. This is also easy to understand: The arbitrageur’s profit is

proportional to the product of the mispricing and the size of his position. The mispricing is

affine in y3S , so the profit is a quadratic function of the arbitrageur’s position, and it is

therefore maximized at a position halfway between zero (which would lead to zero profit)

and a perfect counterparty to the investors (which would lead to zero mispricing, and

hence zero profit).

In contrast to the constant competitive case, the noncompetitive arbitrageur’s stock

position is stochastic, driven by the investors’ difference of opinion ¯ m, the investors’

aggregate wealth W , and the distribution of wealth l. For sufficiently low investor

disagreement, the noncompetitive arbitrageur’s stock position is lower than in the

competitive case ðy3S ðtÞoM Þ, and his position is higher for high investor disagreement.

Hence, for low investor disagreement, the noncompetitive equilibrium mispricing is higher

than the competitive one. Consequently, the economic role of the arbitrageur (as discussed

in Section 4), in terms of alleviating the effect of portfolio constraints and facilitating trade

amongst investors, is reduced. The opposite holds when there is high disagreement

amongst investors: The economic role of the noncompetitive arbitrageur is morepronounced. Nonetheless, our discussion of the economic role of the arbitrageur in

Section 4 remains valid when the exogenous bound on the arbitrageur’s position M is

replaced with the endogenous y3S . Finally, in contrast to the linear competitive case, the

noncompetitive arbitrageur’s profits are convex in the amount of disagreement amongst

investors.

6. Equilibrium with an arbitrageur subject to margin requirements

We now return to a competitive market and assume that the arbitrageur is subject to

margin requirements. The arbitrageur then needs to be endowed with some capital, whichwe assume is held as an investment by the investors. In addition to being more realistic, this

modification of our model allows us to endogenize the amount of capital allocated to

arbitrage activity and the size of the arbitrage positions.

6.1. The arbitrageur under margin requirements

The economic setup introduced in Sections 3 and 4 is modified as follows. Riskless,

costless storage, henceforth referred to as cash, is available to the agents in addition to the

stock, derivative, and bond. Assuming rðtÞ40, the investors would never find it optimal to

invest in cash. This assumption is verified to hold in equilibrium for an appropriate choiceof the exogenous parameters, including the pessimistic investor’s prior belief distribution

on mS . However, the arbitrageur may be forced to hold cash due to his margin

requirements, that are as follows: letting y3C and W 3 denote the arbitrageur’s cash holding


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and capital, respectively, his holdings must obey

W 3ðtÞXZðyS ðtÞ þ jy3P ðtÞjÞ (58)

and

y3C ðtÞXð1 þ ZÞ maxfÀy3

P ðtÞ; 0g, (59)

where Z; 2 ½0; 1 and y3S ðtÞX0 (hence, there is no need to account for short positions in S

in (58)–(59)). While Eq. (58) limits the size of the arbitrageur’s position in proportion to his

capital, Eq. (59) states that the arbitrageur does not enjoy the use of proceeds from his

short position in the derivative; rather, these must be kept in cash as a deposit. In addition,

a fraction of the margin on the short sale must also be met with cash. For clarity, we refer

below to (58) as the ‘‘margin constraint’’ and to (59) as the ‘‘cash constraint.’’ Our

modeling of margin requirements is standard (see, e.g., Cuoco and Liu, 2000). This

modeling of margin requirements could easily be amended without affecting our main

intuition. All that is really needed is the presence of a constraint that limits the size of the

arbitrageur’s position in proportion to his capital, and that short sales be costly. For

simplicity, we assume that there is no difference between initial and maintenance margins.

Denoting y3 y3B ; y

3S ; y

3P ; y

3C

À Á, the arbitrageur’s problem is

maxy3

E 3Z T

0

C3ðtÞ dt

!(60)

s.t. C3ðtÞ dt ¼ fy3S ðtÞm3


P ðtÞ þ y3B ðtÞrðtÞg dt

þ ½y

3

S ðtÞ þ y

3

P ðtÞsdw

3

ðtÞ, ð61Þ

y3S ðtÞ þ y3

P ðtÞ þ y3B ðtÞ þ y3

C ðtÞ ¼ W 3ðtÞ, (62)

ðy3S ðtÞ þ y3

P ðtÞÞ s ¼ 0, (63)

and ð58Þ À ð59Þ hold.

Both the amount of arbitrage capital ðW 3Þ and its return ðC3ðtÞ=W 3ðtÞÞ are determined

endogenously in equilibrium. Restrictions (62) and (63) and the equality in (59) (which

always holds because cash is dominated by the bond) yield all holdings as a function of

y3P ðtÞ:

y3S ðtÞ ¼ Ày3

P ðtÞ; y3B ðtÞ ¼ Àð1 þ ZÞ maxfÀy3

P ðtÞ; 0g, (64)

and

y3C ðtÞ ¼ ð1 þ ZÞ maxfÀy3

P ðtÞ; 0g. (65)

When there is no mispricing ðDS ;P ðtÞ ¼ 0Þ, the arbitrageur is indifferent between all portfolios

satisfying (58), (64)–(65), and y3P ðtÞX0; the last inequality holds because (59) penalizes short

sales (the arbitrageur loses the interest on his cash deposit). As before, all feasible portfolio

holdings yield zero profit. In the presence of mispricing ðDS ;P ðtÞ40Þ, it is optimal for the

arbitrageur to take on as large a profitable arbitrage position (long in the cheap stock, and

short in the more expensive derivative) as feasible; hence, (58) holds with equality:

Zðy3S ðtÞ À y3

P ðtÞÞ ¼ W 3ðtÞ, (66)


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distribution of wealth dynamics, and aggregate wealth dynamics are


1 À lðtÞÀgs2

lðtÞÀ

W 3ðtÞs2

lðtÞð1 À lðtÞÞW ðtÞ

lðtÞð1 þ ZÞ þ 1

2Z

40, (74)

rðtÞ ¼ m2S ðtÞ À ð1 À gÞs2 þ

ðb À 1ÞlðtÞ

1 À lðtÞþ

W 3ðtÞ

ð1 À lðtÞÞW ðtÞ

2 þ Z

2Z

!s2, (75)

dlðtÞ ¼ fDS ;P ðtÞðb À lðtÞÞ þ ðs þ slðtÞÞ½sð1 À 2lðtÞÞ þ slðtÞ þ lðtÞs2g dt

þ lðtÞslðtÞ dw1ðtÞ, ð76Þ

where

slðtÞ ¼ ð1 À lðtÞÞð ¯ mðtÞ À DS ;P ðtÞ=sÞ ÀW 3ðtÞð1 þ ZÞs

2ZW ðtÞ

, (77)

and

dW ðtÞ ¼ W ðtÞ À ð1 þ ZÞ gð1 À lðtÞÞW ðtÞ þW 3ðtÞ

2Z

!& 'ðm1

S ðtÞ dt þ sdw1ðtÞÞ

ÀW ðtÞ

T À tdt. ð78Þ

Because arbitrage is riskless, equilibrium is only possible if the rate of return on

arbitrage capital is equal to the riskless interest rate; otherwise, investors would face an

unbounded arbitrage opportunity and equilibrium would not obtain. From (69), thisimplies that DS ;P ðtÞ ¼ rðtÞð1 þ ZÞ, which leads (using (74) and (75)) to an equation affine in

W 3ðtÞ whose solution is given by (73). The endogenous amount of arbitrage capital W 3 is

provided by Eq. (73). Interestingly, the main factor that drives the equilibrium amount of

arbitrage activity is similar to the noncompetitive case of Section 5: The amount of

arbitrage is driven by the mispricing that would prevail without an arbitrageur,

DIIS ;P ¼ ¯ ms À ðb À 1Þs2=ð1 À lÞ À gs2=l, i.e., the amount of heterogeneity that cannot be

traded by the investors due to their constraints. This is intuitive: the higher this untraded

heterogeneity, the higher the profit opportunities for the arbitrageur/financial inter-

mediary. In addition, the amount of arbitrage capital is increasing in the severity of the

margin constraint (58) (as measured by Z): the more stringent the constraint, the higherthe amount of capital needed to achieve a similar result. In contrast, W 3 is decreasing in

the severity of the cash constraint (59) (i.e., in ð1 þ ZÞ). The rationale for this is clarified by

our discussion on the respective impact of these two constraints on risk-sharing.

The structure of equilibrium closely resembles that of Section 4, with the exogenous

position size M replaced by the endogenous arbitrageur’s stock investment y3S ¼ W 3=2Z.

Nonetheless, risk-sharing between investors is further impacted by the cash constraint.

This is evidenced by the diffusion of the wealth distribution dynamics, sl, which exhibits

an additional term proportional to ð1 þ ZÞ. The expressions for the interest rate and the

mispricing include a similar term, suggesting that the more severe the cash constraint, the

better risk-sharing between investors.To investigate risk-sharing between investors, we examine the investors’ risk exposures,

as provided in Proposition 6.2. We note that the expressions in the proposition are not

dependent of our assumption of logarithmic utility for the investors, as the expressions


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follow from the investors’ binding constraints, the arbitrageur’s policy (67)–(68), and

market clearing. (They would also hold in a pure exchange economy.)

Proposition 6.2. In the mispriced equilibrium with an arbitrageur subject to the margin

requirements (58)–(59), the investors’ risk exposures are

F1ðtÞ ¼ b þ g1 À lðtÞ

lðtÞþ

W 3ðtÞ

2ZlðtÞW ðtÞ, (79)

and

F2ðtÞ ¼ ð1 À gÞ À ðb À 1ÞlðtÞ

1 À lðtÞÀ

ð2 þ ZÞW 3ðtÞ

2Zð1 À lðtÞÞW ðtÞ. (80)

Eq. (80) reveals that the pessimistic investor’s equilibrium risk exposure is decreased by

the severity of the cash constraint (measured by ), which leads to an improvement in risk-

sharing due to the cash constraint. This is because the cash deposit is taken out of theaggregate amount of the good to be invested in production. Hence, market clearing

requires y1S þ y2

S þ y3S ¼ W 3 À y3

C . Thus, the higher the arbitrageur’s cash holding, the less

the pessimistic investor has to invest in the stock for markets to clear. (Investor 2’s

investment has to adjust to clear the markets because investor 1’s leverage constraint on

the stock is binding.) This makes it possible for the pessimistic investor to have a lower risk

exposure, closer to the value it would have in an unconstrained equilibrium. Investor 1’s

risk exposure is unaffected by the arbitrageur’s cash constraint, thereby leading to a higher

value for our measure of risk-sharing ðF1 À F2Þ. Thus, the cash constraint mitigates the

effect of the margin constraint, albeit in an indirect way. From (80), the effect of the cash

constraint on investor 2’s risk exposure is equivalent to an increase in the amount of

arbitrage activity. The effect of the margin constraint (58), on the other hand, is as would

be expected: It is similar to that of the arbitrageur’s investment restriction in Section 4, and

the intuition therein follows. In fact, in the absence of any cash requirement ð1 þ Z ¼ 0Þ,

the expressions from the mispriced equilibrium in Section 4 apply once M is replaced with

the endogenous y3S .

The model of equilibrium arbitrage activity in this section is admittedly simplistic in

many respects. In particular, our implication that the return on arbitrage capital equals the

riskless interest rate is undoubtedly unrealistic. Nonetheless, we believe that the general

approach of this section is consistent with the nature of arbitrage in contemporaryfinancial markets, with specialized agents engaging in arbitrage, such as hedge funds,

whose capital is held by outside investors, and could be extended to a more realistic setup.

Finally, we demonstrate the different ways in which the constraints imposed on the

arbitrageur can affect risk-sharing.

7. Conclusion

In this article, we attempt to shed some light on the potential role of arbitrageurs in

rational financial markets. Towards that end, we develop an economic setting with

heterogeneous risk-averse investors subject to leverage and short-sales restrictions, and anarbitrageur engaging in costless, riskless arbitrage trades. In the presence of arbitrage

opportunities, the arbitrageur is shown to improve risk-sharing amongst investors. The

arbitrageur effectively plays the role of a financial intermediary, or an innovator


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synthesizing a derivative security specifically designed to relieve investors’ restrictions. The

improved transfer of risk due to the arbitrageur is shown to be equally valid when the

arbitrageur behaves noncompetitively or is subject to margin requirements and needs

capital to implement his arbitrage trades. Although we make simplifying assumptions on

the primitives of the economy (preferences, investment opportunities, and investmentrestrictions) for tractability, our insights can be readily extended to more general primitives

and to a pure exchange economy.

Our framework is amenable to the study of further related issues. For example, the

analysis of the arbitrageur under margin requirements could be extended to make arbitrage

activity risky by allowing for the possibility of volatility jumps in a discretized version of

the model. Then, the risk-return tradeoff of arbitrage capital being consistent with asset

prices would pin down the endogenous amount of arbitrage capital. Our model also sheds

new light on the issue of who the users of derivatives markets such as futures are, what the

motivations of the different categories of market participants are, and what returns each

participant type obtains. Typically, the empirical literature on futures markets (e.g.,

Ederington and Lee, 2002; Piazzesi and Swanson, 2004), distinguishes between hedgers and

speculators. Our model underscores the need to additionally account for the activity of

arbitrageurs, and provides theoretical tools to understand how their positions and their

returns are related to underlying economic variables.

Appendix A. Proofs

Proof of Proposition 3.1. The investors’ optimization problem is nonstandard in several

respects: the portfolio constraints, the redundancy in the risky investment opportunities(technology and stock), and the mispricing. The solution technique, developed by Basak

and Croitoru (2000), involves using investors’ nonsatiation (implying that, in the presence

of mispricing, the investment restrictions always bind) to convert the original problem with

two risky investment opportunities into one with a single, fictitious risky asset with

nonlinear drift. Techniques of optimization in nonlinear markets (Cvitanic and Karatzas,

1992) can then be applied. &

Proof of Proposition 4.1. Combining the agents’ optimal policies (Proposition 3.1 and

Section 4.1) and the conditions for market clearing, and noting that both investors face the

same price process for the derivative P , implies

m1P ðtÞ À m2

P ðtÞ

sdt ¼ dw2ðtÞ À dw1ðtÞ ¼

m1S ðtÞ À m2

S ðtÞ

sdt; hence

m1P ðtÞ À m2

P ðtÞ

s¼ ¯ mðtÞ, ðA:1Þ

and leads to the equilibrium expressions for r and DS ;P in each case. Substituting these

equilibrium prices in the conditions for the investors’ optimization cases leads to the

conditions for the equilibrium case ðc; eÞ and the first condition for ða; bÞ. The conditions

for ða; aÞ and the second condition for ða; bÞ are those under which it is possible to find

portfolio holdings that, among all those between which agents are indifferent, clearmarkets.

To verify the conditions for the ða; aÞ case, observe that if the economy is in

ða; aÞ, each of the three agents is indifferent among all the portfolio holdings that


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satisfy, respectively,

y1S ðtÞ

W 1ðtÞþ

y1P ðtÞ

W 1ðtÞ¼m1

S ðtÞ À rðtÞ

s2¼ 1 þ

¯ mðtÞð1 À lðtÞÞ

s, (A.2)

y2S ðtÞ

W 2ðtÞþ

y2P ðtÞ

W 2ðtÞ¼m2

S ðtÞ À rðtÞ

s2¼ 1 À

¯ mðtÞlðtÞ

s, (A.3)

y3P ðtÞ ¼ Ày3

S ðtÞ and y3B ðtÞ ¼ 0, (A.4)

where the equilibrium value for the interest rate has been substituted into the investors’

portfolio demands. In addition, agents’ portfolio holdings must satisfy the clearing

conditions that are equivalent to

y1

S ðtÞ þ y2

S ðtÞ þ y3

S ðtÞ ¼ W 1

ðtÞ þ W 2

ðtÞ; and y1

P ðtÞ þ y2

P ðtÞ þ y3

P ðtÞ ¼ 0. (A.5)

There exist an infinity of solutions to the system formed by Eqs. (A.2)–(A.5). The

conditions for case ða; aÞ provided in the proposition ensure that there exists a solution that

satisfies the agents’ position limits (11) and (23). To verify this, express the solutions of the

system as a function of two free parameters (say, y1P ðtÞ and y3

S ðtÞ), since there are two

degrees of freedom in the system, and substitute these expressions into the position limits.

This results in eight inequalities that y1P ðtÞ and y3

S ðtÞ must obey. The conditions for case

ða; aÞ ensure that there is no contradiction among these, so that there exist portfolio

holdings that satisfy all the requirements for the unconstrained equilibrium ða; aÞ. The

second condition for case ða;

bÞ can be obtained in a similar fashion.

It is easy to verify that no other cases are possible in equilibrium (the only other cases

compatible with market clearing, (b, a) and (e, c), are impossible given that investor 1 is

more optimistic). Substituting the prices and optimal policies into the investors’ dynamic

budget constraints and using Ito’s Lemma leads to the dynamics of l and W , while the

expressions for the investors’ consumptions follow from Proposition 3.1. &

Proof of Corollary 4.1. The expressions follow from the definition of Fi , the optimal

policies in Proposition 3.1, and the equilibrium prices of Proposition 4.1. &

Proof of Proposition 4.2. The expressions follow from the optimal policies in Proposition

3.1 and market clearing. Substituting the equilibrium prices into the conditions for theoptimality cases leads to the conditions for the ðc; e; cÞ equilibrium. Details are omitted for

brevity. &

Proof of Proposition 5.1. In the presence of an active arbitrageur, market clearing and

agents’ optimal policies lead to the expressions for the equilibrium mispricing as a function

of the arbitrageur’s position (50). Substitution into the arbitrageur’s optimization problem

(51) leads to a quadratic problem whose solution is provided by (52). Substitution into the

problem’s objective function leads to the optimal value of C3ðtÞ. Given the arbitrageur’s

position, the equilibrium values of the mispricing, interest rate, distribution of wealth

dynamics, and aggregate wealth dynamics are deduced in a fashion similar to thecompetitive case (Proposition 4.1). The conditions for the equilibrium are those under

which there exists a solution to the arbitrageur’s optimization such that he realizes a

positive profit, and the investors are optimally in case (c, e). &


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Proof of Proposition 6.1. The interest rate and mispricing, as a function of W 3, follow

from the arbitrageur’s holdings in (67)–(68), the investors’ demands (Proposition 3.1), and

market clearing in a fashion similar to Proposition 4.1. To pin down the amount of

arbitrage capital W 3, observe that there only exists a solution to the investors’

optimization if the (riskless) return on arbitrage (69) equals the riskless rate r; otherwise,the investors would face an unbounded arbitrage opportunity. By substituting the interest

rate and mispricing into the expression for the return on arbitrage capital (69), we obtain

(73), the only amount of arbitrage capital such that this restriction holds. The conditions in

the proposition ensure that the aggregate amount invested in arbitrage activity is

nonnegative, and that the investors, given the trades of the arbitrageur, are in cases (c)

and (e). &

Proof of Proposition 6.2. The investors’ portfolio holdings can be deduced from their

binding constraints, the arbitrageur’s holdings (67)–(68), and market clearing. Applying

the definition of Fi then yields the expressions in the proposition. &

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